C₂×#7708431 #72331703

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#7708431

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

71605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

(3,4,7,5)(6,8,10,9)(11,12,14,16)(13,15,17,18)(19,20,23,24)(21,22,25,26);(4,5)(8,9)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,25)(24,26);(3,4,7,5)(6,8,10,9)(11,12)(13,15)(14,16)(17,18)(19,20,23,24)(21,22,25,26);(1,2);(4,5)(6,10)(13,17)(15,18)(20,24)(22,26);(3,5,7,4)(6,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,20)(21,26)(22,25)(23,24);(3,6,7,10)(4,8,5,9)(11,13,14,17)(12,15,16,18)(19,22,23,26)(20,25,24,21)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227
maximal normal subgroup of	C₂×#72241227
maximal quotient of	C₂×#72297705

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4,7,5)(6,8,10,9)(11,12,14,16)(13,15,17,18)(19,20,23,24)(21,22,25,26), (4,5)(8,9)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,25)(24,26), (3,4,7,5)(6,8,10,9)(11,12)(13,15)(14,16)(17,18)(19,20,23,24)(21,22,25,26), (1,2), (4,5)(6,10)(13,17)(15,18)(20,24)(22,26), (3,5,7,4)(6,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,20)(21,26)(22,25)(23,24), (3,6,7,10)(4,8,5,9)(11,13,14,17)(12,15,16,18)(19,22,23,26)(20,25,24,21));

C2×#7708431 #72331703